Problem: Alexandra has a stand in the marketplace where she sells ground cumin. She has a fixed amount of weekly expenses, and she sells her cumin for $\$7.20$ per kilogram. If she sells $90$ kilograms of cumin, her weekly net profit is $\$414$. What are Alexandra's weekly expenses? $\$$
Each kilogram of Alexandra's cumin costs $\$7.20$, so Alexandra makes a profit of $7.2W$ dollars for selling $W$ kilograms of cumin. Alexandra's weekly net profit is found by taking her income from selling cumin and subtracting from it her weekly expenses. We can express this with the equation $P=7.2W-E$, where: $P$ represents Alexandra's weekly net profit (in dollars) $W$ represents the amount of cumin she sold (in kilograms) $E$ represents Alexandra's expenses (in dollars) We want to find $E$, so let's first solve the equation for $E$ : $ \begin{aligned}P&=7.2W-E\\ E&=7.2W-P\end{aligned}$ Now, we know that if Alexandra sells $90$ kilograms of cumin $(W={90})$, her weekly net profit is $\$414$ $(P={414})$. Let's plug these values into the equation to find the value of $E$. $ E=7.2\cdot{90}-{414}=234$ Therefore, Alexandra's weekly expenses are $\$234$. To find how much cumin Alexandra has to sell to cover her weekly expenses, we can plug $P=0$ into the equation and solve for $W$. $ \begin{aligned}234&=7.2W-0\\ 7.2W&=234\\ W&=32.5\end{aligned}$ Alexandra's weekly expenses are $\$234$. Alexandra has to sell $32.5$ kilograms of cumin to cover her weekly expenses.